Theory Field_ZF

theory Field_ZF
imports Ring_ZF
(*
This file is a part of IsarMathLib -
a library of formalized mathematics for Isabelle/Isar.

Copyright (C) 2005, 2006 Slawomir Kolodynski

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*)

header{*\isaheader{Field\_ZF.thy}*}

theory Field_ZF imports Ring_ZF

begin

text{*This theory covers basic facts about fields.*}

section{*Definition and basic properties*}

text{*In this section we define what is a field and list the basic properties
of fields. *}


text{*Field is a notrivial commutative ring such that all
non-zero elements have an inverse. We define the notion of being a field as
a statement about three sets. The first set, denoted @{text "K"} is the
carrier of the field. The second set, denoted @{text "A"} represents the
additive operation on @{text "K"} (recall that in ZF set theory functions
are sets). The third set @{text "M"} represents the multiplicative operation
on @{text "K"}.*}


definition
"IsAfield(K,A,M) ≡
(IsAring(K,A,M) ∧ (M {is commutative on} K) ∧
TheNeutralElement(K,A) ≠ TheNeutralElement(K,M) ∧
(∀a∈K. a≠TheNeutralElement(K,A)-->
(∃b∈K. M`⟨a,b⟩ = TheNeutralElement(K,M))))"


text{*The @{text "field0"} context extends the @{text "ring0"}
context adding field-related assumptions and notation related to the
multiplicative inverse. *}


locale field0 = ring0 K A M for K A M +
assumes mult_commute: "M {is commutative on} K"

assumes not_triv: "\<zero> ≠ \<one>"

assumes inv_exists: "∀a∈K. a≠\<zero> --> (∃b∈K. a·b = \<one>)"

fixes non_zero ("K0")
defines non_zero_def[simp]: "K0 ≡ K-{\<zero>}"

fixes inv ("_¯ " [96] 97)
defines inv_def[simp]: "a¯ ≡ GroupInv(K0,restrict(M,K0×K0))`(a)";

text{*The next lemma assures us that we are talking fields
in the @{text "field0"} context.*}


lemma (in field0) Field_ZF_1_L1: shows "IsAfield(K,A,M)"
using ringAssum mult_commute not_triv inv_exists IsAfield_def
by simp;

text{*We can use theorems proven in the @{text "field0"} context whenever we
talk about a field.*}


lemma field_field0: assumes "IsAfield(K,A,M)"
shows "field0(K,A,M)"
using assms IsAfield_def field0_axioms.intro ring0_def field0_def
by simp;

text{*Let's have an explicit statement that the multiplication
in fields is commutative.*}


lemma (in field0) field_mult_comm: assumes "a∈K" "b∈K"
shows "a·b = b·a"
using mult_commute assms IsCommutative_def by simp;

text{*Fields do not have zero divisors.*}

lemma (in field0) field_has_no_zero_divs: shows "HasNoZeroDivs(K,A,M)"
proof -
{ fix a b assume A1: "a∈K" "b∈K" and A2: "a·b = \<zero>" and A3: "b≠\<zero>"
from inv_exists A1 A3 obtain c where I: "c∈K" and II: "b·c = \<one>"
by auto;
from A2 have "a·b·c = \<zero>·c" by simp;
with A1 I have "a·(b·c) = \<zero>"
using Ring_ZF_1_L11 Ring_ZF_1_L6 by simp
with A1 II have "a=\<zero> "using Ring_ZF_1_L3 by simp }
then have "∀a∈K.∀b∈K. a·b = \<zero> --> a=\<zero> ∨ b=\<zero>" by auto;
then show ?thesis using HasNoZeroDivs_def by auto;
qed;

text{*$K_0$ (the set of nonzero field elements is closed with respect
to multiplication.*}


lemma (in field0) Field_ZF_1_L2:
shows "K0 {is closed under} M"
using Ring_ZF_1_L4 field_has_no_zero_divs Ring_ZF_1_L12
IsOpClosed_def by auto;

text{*Any nonzero element has a right inverse that is nonzero.*}

lemma (in field0) Field_ZF_1_L3: assumes A1: "a∈K0"
shows "∃b∈K0. a·b = \<one>"
proof -
from inv_exists A1 obtain b where "b∈K" and "a·b = \<one>"
by auto;
with not_triv A1 show "∃b∈K0. a·b = \<one>"
using Ring_ZF_1_L6 by auto;
qed;

text{*If we remove zero, the field with multiplication
becomes a group and we can use all theorems proven in
@{text "group0"} context.*}


theorem (in field0) Field_ZF_1_L4: shows
"IsAgroup(K0,restrict(M,K0×K0))"
"group0(K0,restrict(M,K0×K0))"
"\<one> = TheNeutralElement(K0,restrict(M,K0×K0))"
proof-
let ?f = "restrict(M,K0×K0)"
have
"M {is associative on} K"
"K0 ⊆ K" "K0 {is closed under} M"
using Field_ZF_1_L1 IsAfield_def IsAring_def IsAgroup_def
IsAmonoid_def Field_ZF_1_L2 by auto;
then have "?f {is associative on} K0"
using func_ZF_4_L3 by simp;
moreover
from not_triv have
I: "\<one>∈K0 ∧ (∀a∈K0. ?f`⟨\<one>,a⟩ = a ∧ ?f`⟨a,\<one>⟩ = a)"
using Ring_ZF_1_L2 Ring_ZF_1_L3 by auto;
then have "∃n∈K0. ∀a∈K0. ?f`⟨n,a⟩ = a ∧ ?f`⟨a,n⟩ = a"
by blast;
ultimately have II: "IsAmonoid(K0,?f)" using IsAmonoid_def
by simp;
then have "monoid0(K0,?f)" using monoid0_def by simp;
moreover note I
ultimately show "\<one> = TheNeutralElement(K0,?f)"
by (rule monoid0.group0_1_L4);
then have "∀a∈K0.∃b∈K0. ?f`⟨a,b⟩ = TheNeutralElement(K0,?f)"
using Field_ZF_1_L3 by auto;
with II show "IsAgroup(K0,?f)" by (rule definition_of_group)
then show "group0(K0,?f)" using group0_def by simp
qed;

text{*The inverse of a nonzero field element is nonzero.*}

lemma (in field0) Field_ZF_1_L5: assumes A1: "a∈K" "a≠\<zero>"
shows "a¯ ∈ K0" "(a¯)2 ∈ K0" "a¯ ∈ K" "a¯ ≠ \<zero>"
proof -
from A1 have "a ∈ K0" by simp;
then show "a¯ ∈ K0" using Field_ZF_1_L4 group0.inverse_in_group
by auto;
then show "(a¯)2 ∈ K0" "a¯ ∈ K" "a¯ ≠ \<zero>"
using Field_ZF_1_L2 IsOpClosed_def by auto
qed;

text{*The inverse is really the inverse.*}

lemma (in field0) Field_ZF_1_L6: assumes A1: "a∈K" "a≠\<zero>"
shows "a·a¯ = \<one>" "a¯·a = \<one>"
proof -
let ?f = "restrict(M,K0×K0)"
from A1 have
"group0(K0,?f)"
"a ∈ K0"
using Field_ZF_1_L4 by auto;
then have
"?f`⟨a,GroupInv(K0, ?f)`(a)⟩ = TheNeutralElement(K0,?f) ∧
?f`⟨GroupInv(K0,?f)`(a),a⟩ = TheNeutralElement(K0, ?f)"

by (rule group0.group0_2_L6);
with A1 show "a·a¯ = \<one>" "a¯·a = \<one>"
using Field_ZF_1_L5 Field_ZF_1_L4 by auto;
qed;

text{*A lemma with two field elements and cancelling.*}

lemma (in field0) Field_ZF_1_L7: assumes "a∈K" "b∈K" "b≠\<zero>"
shows
"a·b·b¯ = a"
"a·b¯·b = a"
using assms Field_ZF_1_L5 Ring_ZF_1_L11 Field_ZF_1_L6 Ring_ZF_1_L3
by auto;

section{*Equations and identities*}

text{*This section deals with more specialized identities that are true in
fields.*}


text{*$a/(a^2) = 1/a$.*}

lemma (in field0) Field_ZF_2_L1: assumes A1: "a∈K" "a≠\<zero>"
shows "a·(a¯)2 = a¯"
proof -
have "a·(a¯)2 = a·(a¯·a¯)" by simp;
also from A1 have "… = (a·a¯)·a¯"
using Field_ZF_1_L5 Ring_ZF_1_L11
by simp;
also from A1 have "… = a¯"
using Field_ZF_1_L6 Field_ZF_1_L5 Ring_ZF_1_L3
by simp;
finally show "a·(a¯)2 = a¯" by simp;
qed;

text{*If we multiply two different numbers by a nonzero number, the results
will be different.*}


lemma (in field0) Field_ZF_2_L2:
assumes "a∈K" "b∈K" "c∈K" "a≠b" "c≠\<zero>"
shows "a·c¯ ≠ b·c¯"
using assms field_has_no_zero_divs Field_ZF_1_L5 Ring_ZF_1_L12B
by simp;

text{*We can put a nonzero factor on the other side of non-identity
(is this the best way to call it?) changing it to the inverse.*}


lemma (in field0) Field_ZF_2_L3:
assumes A1: "a∈K" "b∈K" "b≠\<zero>" "c∈K" and A2: "a·b ≠ c"
shows "a ≠ c·b¯"
proof -
from A1 A2 have "a·b·b¯ ≠ c·b¯"
using Ring_ZF_1_L4 Field_ZF_2_L2 by simp;
with A1 show "a ≠ c·b¯" using Field_ZF_1_L7
by simp;
qed;

text{*If if the inverse of $b$ is different than $a$, then the
inverse of $a$ is different than $b$.*}


lemma (in field0) Field_ZF_2_L4:
assumes "a∈K" "a≠\<zero>" and "b¯ ≠ a"
shows "a¯ ≠ b"
using assms Field_ZF_1_L4 group0.group0_2_L11B
by simp;

text{*An identity with two field elements, one and an inverse.*}

lemma (in field0) Field_ZF_2_L5:
assumes "a∈K" "b∈K" "b≠\<zero>"
shows "(\<one> \<ra> a·b)·b¯ = a \<ra> b¯"
using assms Ring_ZF_1_L4 Field_ZF_1_L5 Ring_ZF_1_L2 ring_oper_distr
Field_ZF_1_L7 Ring_ZF_1_L3 by simp;

text{*An identity with three field elements, inverse and cancelling.*}

lemma (in field0) Field_ZF_2_L6: assumes A1: "a∈K" "b∈K" "b≠\<zero>" "c∈K"
shows "a·b·(c·b¯) = a·c"
proof -
from A1 have T: "a·b ∈ K" "b¯ ∈ K"
using Ring_ZF_1_L4 Field_ZF_1_L5 by auto;
with mult_commute A1 have "a·b·(c·b¯) = a·b·(b¯·c)"
using IsCommutative_def by simp;
moreover
from A1 T have "a·b ∈ K" "b¯ ∈ K" "c∈K"
by auto;
then have "a·b·b¯·c = a·b·(b¯·c)"
by (rule Ring_ZF_1_L11);
ultimately have "a·b·(c·b¯) = a·b·b¯·c" by simp;
with A1 show "a·b·(c·b¯) = a·c"
using Field_ZF_1_L7 by simp;
qed

section{*1/0=0*}

text{* In ZF if $f: X\rightarrow Y$ and $x\notin X$ we have $f(x)=\emptyset$.
Since $\emptyset$ (the empty set) in ZF is the same as zero of natural numbers we
can claim that $1/0=0$ in certain sense. In this section we prove a theorem that
makes makes it explicit.*}


text{*The next locale extends the @{text "field0"} locale to introduce notation
for division operation.*}


locale fieldd = field0 +
fixes division
defines division_def[simp]: "division ≡ {⟨p,fst(p)·snd(p)¯⟩. p∈K×K0}"

fixes fdiv (infixl "\<fd>" 95)
defines fdiv_def[simp]: "x\<fd>y ≡ division`⟨x,y⟩"


text{*Division is a function on $K\times K_0$ with values in $K$.*}

lemma (in fieldd) div_fun: shows "division: K×K0 -> K"
proof -
have "∀p ∈ K×K0. fst(p)·snd(p)¯ ∈ K"
proof
fix p assume "p ∈ K×K0"
hence "fst(p) ∈ K" and "snd(p) ∈ K0" by auto
then show "fst(p)·snd(p)¯ ∈ K" using Ring_ZF_1_L4 Field_ZF_1_L5 by auto
qed
then have "{⟨p,fst(p)·snd(p)¯⟩. p∈K×K0}: K×K0 -> K"
by (rule ZF_fun_from_total)
thus ?thesis by simp
qed

text{*So, really $1/0=0$. The essential lemma is @{text "apply_0"} from standard
Isabelle's @{text "func.thy"}.*}


theorem (in fieldd) one_over_zero: shows "\<one>\<fd>\<zero> = 0"
proof-
have "domain(division) = K×K0" using div_fun func1_1_L1
by simp
hence "⟨\<one>,\<zero>⟩ ∉ domain(division)" by auto
then show ?thesis using apply_0 by simp
qed

end